Atlas › Modular forms › Weight 2 › Level 589 › 589.2.a › 589.2.a.b

Newform orbit 589.2.a.b

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q - 4q^{2} - 5q^{3} + 8q^{4} - 10q^{5} - q^{6} - q^{7} - 6q^{8} + 4q^{9} + 2q^{10} - 17q^{11} - 3q^{13} - 11q^{14} - 2q^{15} - 10q^{16} - 5q^{17} - 10q^{18} + 9q^{19} - 2q^{20} - 2q^{21} - 10q^{22} - 3q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{9} + 9x^{8} + 13x^{7} - 75x^{6} - 166x^{5} + 209x^{4} + 423x^{3} - 255x^{2} - 44x + 8$

Atkin–Lehner signs

$p$	$w_p$
19	$-1$
31	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	−4	−5	−10	−1	−17	−3	−5	9	−3	−26	9	−5	−29	3	−12	−14	−47	13	6	−41	11	−3	−17	−24	8

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.201 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 19, 31, 589]: [0, 2, 3, 54] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(589, 2)[1]   # 589.2.a.b

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 589:2 1000